Directly ahead of a mode I crack tip in a thick section of a ductile alloy

PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Solid continuum mechanics and complex fluids · Solid continuum mechanics and complex fluids

Bending, Ringing, Breaking and Flowing, Ch Fracture: stress concentration, cracks and toughness

Problem

Directly ahead of a mode I crack tip in a thick section of a ductile alloy, the near-tip stresses on the crack plane are σxx=σyy=σ\sigma_{xx} = \sigma_{yy} = \sigma and, because lateral contraction is prevented (plane strain), σzz=2νσ\sigma_{zz} = 2\nu\sigma, with Poisson's ratio ν=1/3\nu = 1/3 for this alloy. The uniaxial yield stress is σY\sigma_Y. ((a)) Using the von Mises yield criterion, find the value of σ\sigma at which yielding first occurs on the crack plane, expressed as a multiple of σY\sigma_Y. ((b)) The first-order plastic-zone estimate caps the crack-tip field σyy≈KI/2πr\sigma_{yy} \approx K_I/\sqrt{2\pi r} at the stress found in part (a), giving rp∝1/(cap stress)2r_p \propto 1/(\text{cap stress})^2 for fixed KIK_I. Find the ratio of the plane-strain plastic-zone size to the plane-stress estimate rp=(KI/σY)2/2πr_p = (K_I/\sigma_Y)^2/2\pi, for the same KIK_I.

Browse the problem library